Eat rook found in the arts arena (7)
I believe the answer is:
theatre
'arts arena' is the definition.
(arena or venue for an arts performance)
'eat rook found in the' is the wordplay.
'rook' becomes 'R' (chess abbreviation).
'found in' means one lot of letters goes inside another.
'eat'+'r'='eatr'
'eatr' going into 'the' is 'THEATRE'.
(Other definitions for theatre that I've seen before include "Part of hospital" , "Playhouse or sphere of action" , "World of drama" , "Operating area" , "Hospital operating room - playhouse" .)